Learning Forum

[Ruzan Sokhoyan]

Hello,

I am running convergence tests for RCWA simulations and trying to find the optimal value for the k value in the RCWA solver. I simulate a silicon pillar on a silica (SiO2) substrate. The FDTD and RCWA simulation results are similar but do not fully match (I am currently running more FDTD simulations with a finer mesh).

When increasing k value in the RCWA I observe a red shift of the resonance. But when increase the k value from 900 to 1400, I observe a blue shift of the resonance. Based on your experience, is it reasonable to have such high values of k? Should I try increasing k values further and further? Or from your perspective this match is good enough?

 

Thanks!

 

[Guilin Sun]

This is an interesting topic. I tend to believe that the shift is due to the RCWA converging accuracy. It fluctuates around the theoretical result. For FDTD, it has numerical dispersion error (and other numerical errors such as PML and source injection etc). So it is hard to say where the theoretical location is before you do converging test.

For RCWA converging testing, it might not be monotonic when you only change one parameter to certain point. 

[Ruzan Sokhoyan]

Changing PML conditions doesn't affect the results of the FDTD simulations. On the other hand, I would expect that reducing mesh would affect the simulation results. The Si pillar is 850 nm-tall, and running it with 4 nm-mesh on 40 cores takes many hours. I am wondering if the change in FDTD result primarily comes from the different geometrical sampling.

1. Is there a straightforward way to ensure that the geometry of the structure does not change when I change the mesh?
2. My understanding is that introducing and external mesh object in the RCWA simulation does not influence the simulation results. Is that correct?
3. In the future, I plan to add thin material layers in my RCWA simulation. These layers can be as thin as 0.8 nm. If my question 2 is correct, how can I ensure that the field is properly resolved in these ultrathin layers?

Thanks!

[Guilin Sun]

A1: I am not sure I understand this question. In FDTD, geometries are discretized. With finer mesh it better represents the geometries. When you set the geometry it does not change, except with better representation with finer mesh.

However, the simulated results are not only dependent on mesh, but also some other settings. Please refer to the FDTD converging page Convergence testing process for FDTD simulations

A2: RCWA meshing is also a factor affecting result accuracy.

A3: in RCWA, the thickness of a geometry does not affect accuracy since the interfaces are specified in RCWA and discretization is in the lateral 2D dimension.

 

[Ruzan Sokhoyan]

Thank you. I have follow-up questions. 

1. Is it possible to estimate how long the simulation will run in RCWA? FDTD gives such an estimation at the beginning of the simulation. What about RCWA?
2. Is it theoretically possible that the RCWA and FDTD simulations never converge in certain situations?
3. I observe that the RCWA simulation crushes for k values above 1800. What are the highest k values you have ever used in your simulations? Have you also observed the "crushing behavior"?

Thank you!

[Guilin Sun]

A1: currently this feature is not in place yet.

A2: no, they can converge to the same result, provided that the seetings are "proper". It just takes time and efforts. Such converging testings have been done with some benchmarks.

A3: When the number of k points is very large, it demands much more resources. When the computer cannot handle then it may crush. I think it is better to crush than freeze otherwise you could not do anything on your computer.

 

 

[Ruzan Sokhoyan]

Hi Guilin,

 

Thank you for your response. 

1. First I checked the convergence of FDTD. for 10 nm mesh, I observed that changing the number of PML layers does not affect the result. Then I reduced the mesh and observed that the simulation results for 3 nm mesh and 5 nm mesh are very close. Based on this I can conclude that FDTD simulation has converged. Do you agree with this?
2. Then I started comparing the RCWA simulation results with FDTD. I change k numbers and also a number of mesh points from within the RCWA solver. I use 40 threads and 100 or 70 wavelength points in RCWA simulation and the simulation runs much longer than FDTD. Still, I haven't been able to achieve convergence. Even for values k=1700, I do not see convergence. When I increase the number of mesh points in RCWA  to 200 (from default 100) the discrepancy between RCWA and FDTD increases. Do you have a piece of specific advice on how to proceed here? Would you suggest keeping indefinitely increasing k? Note that in RCWA, the number of mesh points 200 and 400 yielded identical result that's why I decided to stick with 200.

Thanks!

 

[Guilin Sun]

In FDTD the peak location may also depend on the simulation time as it accumulates the power for each round trip of the light. You can use smaller autoshutoff min with sufficiently long simulation time. you can use check point for continuing simulation with smaller autoshut off min:https://optics.ansys.com/hc/en-us/articles/360034382534-FDTD-solver-Simulation-Object

When you say PML does not affect the result, it might be true for your specific mesh. When mesh becomes finer please make sure PML has sufficient thickness.

RCWA converging test is a little tricky. Please enable "use tangent vector field". Make sure the mesh refinement uses the same as FDTD's. Use larger difference of k vectors, say 500,1000,1500 to see the trends. When too many k vector values it can be slow You know you are doing the converging test, not direct simulation with resonable result. RCWA uses Fourier decomposition of the complex fields. So if there is discontinuous, it will need many Fourier waves. Please refer to Wiki on RCWA to get more info, eg, efficency, memory and time.

 

[Ruzan Sokhoyan]

Hello,

I would also like to provide further information regarding convergence tests. Previoulsy I tried to simulate a high-Q mode supported by the structure. Now, I look into a lower-Q mode supported by the structure, and I have been able to reach the convergence conditions for the RCWA simulations. For this lower-Q mode k=1600 and k=2000 yield identical results. However, these do not match the converged results of the FDTD simulations (see attached images). Is it possible that this is the best achievebale matching between RCWA and FDTD and it is not possible to improve the result further?

 

 

[Ruzan Sokhoyan]

Hi Guilin,

 

Thank you for your suggestion to use a lower auto-shutoff. I did use lower auto-shutoff with 10 nm mesh and saw no difference. Right now I use 10^-7. But 10^-8 yielded the same result. I also changed PML from the default 8 layers to 64 when I had 5 nm mesh, and I observed no difference.

 

Regarding RCWA, I use "tangent vector field" setting.

 

 

[Ruzan Sokhoyan]

I ran a number of RCWA simulations trying to reproduce the high-Q mode at a wavelength of ~1530 nm. I fixed the wavelength in these simulations since each simulation runs for a few hours. for k=3640 I got a transmittance of 0.0988, for k=3820, I got a transmittance of 0.064, for k=4000, I got a transmittance of 0.084. 

1. Is it fair to conclude that RCWA is not able to provide accurate results for high-Q modes?
2. What was the highest k-value you used in your benchmark simulations?

Thank you!

[Guilin Sun]

Sorry just noticed your replies as the system stops sending reminder information.

If you do more details convergence testing, both RCWA and FDTD will reach the same result, in theory. In practice, there are different factors that affects the accuracy. What is more, the time and efforts you spend on it might not be proportional to the improvements of the results. As you showed, autoshutoff min 1e-7 and 1e-8 reaches the "same" result. did you use more finer mesh and thicker PML? in both cases are they terminated with the autoshutoff level or the simulation time? please get more in formaiton on FDTD converging testing: 

1.      Convergence testing process for FDTD simulations 
2.  

When the mesh reaches certain level, other factors may dominate the errors, till to some point. 

Both RCWA and FDTD solves the Maxwell's Equations however in different ways. There is nothing fundamentally block one solver to reach more accurate result, for whatever the Q values. As you may know, RCWA uses the Fourier modes analytically to represent the complicated fields. In principle the more K values the more accurate, provided that mesh and material (fitting) are proper. I do not find any limit for the maximal k-values except for the computer resources limit.

When you cannot see the difference, you may notice some when you use log plot.